A350219 Decimal expansion of 16*(Pi-1) / (5*Pi^2 - 4*(Pi-1)): an approximation for sin(1) from Bhāskara I's sine approximation formula.

8, 4, 0, 2, 1, 8, 1, 1, 9, 8, 8, 0, 3, 7, 9, 2, 1, 5, 4, 6, 1, 6, 0, 8, 3, 2, 5, 6, 7, 7, 2, 4, 4, 6, 9, 8, 2, 9, 7, 9, 4, 1, 0, 9, 5, 6, 9, 1, 4, 7, 1, 5, 4, 3, 2, 4, 3, 0, 2, 8, 5, 9, 7, 5, 6, 2, 2, 4, 4, 6, 1, 3, 9, 8, 6, 0, 0, 9, 1, 5, 3, 8, 2, 3, 8, 3, 0, 2, 4, 2, 6, 5, 1, 7

Offset

0,1

Comments

The Indian mathematician Bhāskara I (c. 600 - c. 680) proposed this remarkable approximation formula for sin(x), in his work Mahabhaskariya, chapter 7:

sin(x) ~ 16*x*(Pi-x) / (5*Pi^2 - 4*x*(Pi-x)), x in radian, 0 <= x <= Pi.

Formula and sine coincide for x = 0, Pi/6, Pi/2, 5Pi/6, and Pi.

Sin(1) = 0.8414... (A049469) while approximation = 0.8402...

Links

Table of n, a(n) for n=0..93.

Wikipedia, Bhāskara I.

Wikipedia, Bhāskara I's sine approximation formula.

Index entries for transcendental numbers

Formula

Equals 16*(Pi-1) / (5*Pi^2 - 4*(Pi-1)).

Example

0.8402181198803792154616083256772446982979410956914...

Maple

evalf(16*(Pi-1) / (5*Pi^2 - 4*(Pi-1)), 100);

Mathematica

RealDigits[16*(Pi - 1)/(5*Pi^2 - 4*(Pi - 1)), 10, 100][[1]] (* Amiram Eldar, Mar 27 2022 *)

Prog

(PARI) 16*(Pi-1)/(5*Pi^2-4*Pi-4) \\ Charles R Greathouse IV, Oct 02 2022

Crossrefs

Cf. A049469.

Sequence in context: A100253 A175040 A238292 * A200108 A088397 A021123

Adjacent sequences: A350216 A350217 A350218 * A350220 A350221 A350222

Keyword

nonn,cons

Author

Bernard Schott, Mar 27 2022

Status

approved